# Why can we only add or subtract things of the same dimension (meters/second, newtons, etc)? [duplicate]

Dimension analysis is a nice tool to create functions using physics dimensions that are desirable for our lives. I know, as an axiom, of sorts to me, that addition and subtraction must conserve units, and thus are actually relatively uncommon in physics equations.

Now, for why that is, I would tell someone it's the same reason why you only can add terms with the same qualities as that term in math as well (such as $x^2$ only being able to be added with scalar multiples of $x^2$, the same applying to $y^5$ or $e^{-2x}\cosh(3x)$, for example). Except in the case of inputs, we have... (and here is where my explanation gets shady), vector units or magnitudes with magnitude 1 of things such as $\text{meters/second}$, $\text{Newtons}$, $\text{Tesla}$, $\text{Joules/Coulomb}$ etc.

However, I can't seem to necessarily explain why I couldn't say, add $x^2$ to $x$ or a velocity to a force other than my body simply not letting me out of habit, and my only explanation is: "You just can't", or "It'd look bizarre."

I need a better explanation for this. Could someone enlighten me? My two questions:

• Why does adding and subtraction break down at things with different dimensions?

• Why does multiplication/division allow for it?

## marked as duplicate by Kyle Kanos, By Symmetry, Emilio Pisanty, John Rennie, Hritik NarayanJun 2 '17 at 17:43

• Multiplication/division can be explained because most of the relation between different quantities are derived from the proportionality or observation, eg. in capacitor $\text{q}\propto\text{V}\implies\text{q}=\text{CV}$. We just keep using them by substituting these quantities to another which consequently make it happen. – Saharsh Jun 2 '17 at 17:27
• In the future, maybe we find a new force $F$ which satisfies the following law $F = \alpha (r + m_1 m_2)$. So, in the future maybe we can add different quantities. – tchappy ha Feb 1 at 2:38
• But, essentially, why is that adding 2 meters to 3 kg gives you "2 meters and 3 kg", but when multiplying you have $6 meters\ kg$? Why can't you have $5 meters\ kg$? Why is this the mechanism? – sangstar Jun 2 '17 at 17:49